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0π 1(Maps)π 2(MapsDiff)B(×)π 1(Maps)π 1(MapsDiff)BMCGπ 0(BDiff) 0 \to \pi_1(Maps) \to \pi_2\big(Maps \sslash Diff\big) \to B (\mathbb{Z} \times \mathbb{Z}) \to \pi_1(Maps) \to \pi_1\big(Maps \sslash Diff\big) \to B MCG \to \mathbb{Z} \to \mathbb{Z} \to \pi_0(B Diff)

Annular braid group

The annular braid group Br n(An)Br_n(An) on nn \in \mathbb{N} strands is the surface braid group of the annulus, hence the fundamental group of the [configuration space of points|configuration space of -points]] inside the annulus.

Properties

Proposition

The annular braid group is isomorphic to a semidirect product

Br n(An)Br n aff Br_n(An) \,\simeq\, Br_n^{aff} \rtimes \mathbb{Z}

of the the affine braid group? with the group of integers, the latter generated by the braid which exhibits a 1-step cyclic permutation.

References

Asymptotic gauge groups

Given a gauge theory (and/or gravity) on a spacetime with asymptotic boundary, certain would-be gauge transformations (diffeomorphisms) that act non-trivially on asymptotic “boundary data” may in fact be identified as physically observable global symmetries and hence have, in contrast to actual gauge symmetries, “direct empirical significance” (DES, Teh 2016).

A key example is (symmetry generated by) the ADM mass and generally the BMS group? of asymptotic symmetries in asymptotically flat spacetimes.

Up to technical fine-print (cf. Borsboom & Posthuma 2015) a group of asymptotic symmetries is the coset space of all gauge symmetries that respect boundary data by the subgroup of bulk gauge transformations which act as the identity map on the asymptotic boundary (cf. Strominger 2018 (2.10.1), Borsboom & Posthuma 2015 p 2):

AsymptoticSymmetries=BoundaryAdmissibleGaugeSymmetriesBulkGaugeSymmetriesTrivialAtBoundary AsymptoticSymmetries \;=\; \frac{ BoundaryAdmissibleGaugeSymmetries }{ BulkGaugeSymmetriesTrivialAtBoundary }

Hence if the bulk gauge symmetries form a normal subgroup then the asymptotic symmetries form a quotient group characterized by a short exact sequence of the form

1BulkGaugeSymmetriesTrivialAtBoundaryBoundaryAdmissibleGaugeSymmetriesAsymptoticSymmetries1. 1 \to BulkGaugeSymmetriesTrivialAtBoundary \longrightarrow BoundaryAdmissibleGaugeSymmetries \longrightarrow AsymptoticSymmetries \to 1 \,.

References

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Definition

By the spherical braid group Br n(S 2)Br_n(S^2), for nn \in \mathbb{N}, one means the surface braid group

Br n(S 2)π 1Conf n(S 2), Br_n(S^2) \;\simeq\; \pi_1 Conf_n(S^2) \,,

where the surface in question is the 2-sphere S 2S^2. Hence the surface braid group is the fundamental group π 1()\pi_1(-) of the configuration space of n n -points, Conf n()Conf_n(-), on the 2-sphere.

Properties

Proposition

The spherical braid group is the quotient group of the ordinary braid group by one further relation:

Br n(S 2)Br n/((b 1b 2b n1)(b n1b 2b 1), Br_n(S^2) \;\simeq\; Br_n/ \big( (b_1 b_2 \cdots b_{n-1})(b_{n-1} \cdots b_2 b_1) \,,

where the b ib_i denote the Artin braid generators.

Moreover, the canonical map from the plain braid group to the symmetric group factors through this quotient map to the spherical braid group

(Fadell & Van Buskirk 1961 p 245, 255, cf. Tan 2024 §3.1)

References

π 2S 2π 1Maps *(S 2,S 2)π 1Maps(S 2,S 2)π 1S 2π 0Maps *(S 2,S 2)π 0Maps(S 2,S 2)π 0S 2 \pi_2 S^2 \to \pi_1 \Maps^\ast(S^2, S^2) \to \pi_1 Maps(S^2, S^2) \to \pi_1 S^2 \to \pi_0 \Maps^\ast(S^2, S^2) \to \pi_0 Maps(S^2, S^2) \to \pi_0 S^2

Example

The unitarization of the standard representation of the symmetric group Sym 3Sym_3 has the two generating transpositions represented by (what in quantum information theory is called)

  1. the Pauli Z-gate ZZ

  2. its composition ZR y(3π/2)- Z \circ R_y(3\pi/2) with rotation gates.

Proof

For definiteness of computation, when group averaging we will be cycling through the elements of Sym 3Sym_3 in this order:

Sym 3={(123), (132), (213), (231), (312), (321)}. Sym_3 \;=\; \left\{ \begin{array}{l} (1 2 3) ,\,\\ (1 3 2) ,\,\\ (2 1 3) ,\,\\ (2 3 1) ,\,\\ (3 1 2) ,\,\\ (3 2 1) \end{array} \right\} \,.

Let {e 1,e 2,e 3}\big\{e_1, e_2, e_3\big\} denote the canonical linear basis of the defining representation, with group action given by

σe ie σ(i). \sigma \cdot e_i \;\coloneqq\; e_{\sigma(i)} \,.

Then a linear basis for the standard representation inside the defining representation is:

[1 0]e 1e 3 [0 1]e 2e 3. \begin{array}{ccc} \left[\begin{array}{c}1 \\ 0\end{array}\right] \;\coloneqq\; e_1 - e_3 \\ \left[\begin{array}{c}0 \\ 1\end{array}\right] \;\coloneqq\; e_2 - e_3 \mathrlap{\,.} \end{array}

The group-averaged inner product of these basis elements is found to be:

[1 0],[0 1]=([1 0][0 1]+[1 1][0 1]+[0 1][1 0]+[1 1][1 0]+[0 1][1 1]+[1 0][1 1])/6=4/6=2/3 \left\langle \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] ,\, \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] \right\rangle \;=\; \left( \;\; \begin{array}{l} \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] \cdot \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] + \left[ \begin{array}{c} 1 \\ -1 \end{array} \right] \cdot \left[ \begin{array}{c} 0 \\ -1 \end{array} \right] + \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] \cdot \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] + \left[ \begin{array}{c} -1 \\ 1 \end{array} \right] \cdot \left[ \begin{array}{c} -1 \\ 0 \end{array} \right] + \left[ \begin{array}{c} 0 \\ -1 \end{array} \right] \cdot \left[ \begin{array}{c} 1 \\ -1 \end{array} \right] + \left[ \begin{array}{c} -1 \\ 0 \end{array} \right] \cdot \left[ \begin{array}{c} -1 \\ 1 \end{array} \right] \end{array} \right)/6 \;=\; 4/6 \;=\; 2/3
[0 1],[0 1]=([0 1][0 1]+[0 1][0 1]+[1 0][1 0]+[1 0][1 0]+[1 1][1 1]+[1 1][1 1])/6=8/6=4/3 \left\langle \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] ,\, \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] \right\rangle \;=\; \left( \;\; \begin{array}{l} \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] \cdot \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] + \left[ \begin{array}{c} 0 \\ -1 \end{array} \right] \cdot \left[ \begin{array}{c} 0 \\ -1 \end{array} \right] + \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] \cdot \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] + \left[ \begin{array}{c} -1 \\ 0 \end{array} \right] \cdot \left[ \begin{array}{c} -1 \\ 0 \end{array} \right] + \left[ \begin{array}{c} 1 \\ -1 \end{array} \right] \cdot \left[ \begin{array}{c} 1 \\ -1 \end{array} \right] + \left[ \begin{array}{c} -1 \\ 1 \end{array} \right] \cdot \left[ \begin{array}{c} -1 \\ 1 \end{array} \right] \end{array} \right)/6 \;=\; 8/6 \;=\; 4/3
[1 0],[1 0]=([1 0][1 0]+[1 1][1 1]+[0 1][0 1]+[1 1][1 1]+[0 1][0 1]+[1 0][1 0])/6=8/6=4/3 \left\langle \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] ,\, \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] \right\rangle \;=\; \left( \;\; \begin{array}{l} \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] \cdot \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] + \left[ \begin{array}{c} 1 \\ -1 \end{array} \right] \cdot \left[ \begin{array}{c} 1 \\ -1 \end{array} \right] + \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] \cdot \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] + \left[ \begin{array}{c} -1 \\ 1 \end{array} \right] \cdot \left[ \begin{array}{c} -1 \\ 1 \end{array} \right] + \left[ \begin{array}{c} 0 \\ -1 \end{array} \right] \cdot \left[ \begin{array}{c} 0 \\ -1 \end{array} \right] + \left[ \begin{array}{c} -1 \\ 0 \end{array} \right] \cdot \left[ \begin{array}{c} -1 \\ 0 \end{array} \right] \end{array} \right)/6 \;=\; 8/6 \;=\; 4/3

From this an orthonormal basis for the averaged inner product is

|012[1 1]=12(e 1+e 2)e 3 |132[1 1]=32(e 1e 2). \begin{array}{l} {\vert 0 \rangle} \;\coloneqq\; \tfrac{1}{2} \left[ \begin{array}{c} 1 \\ 1 \end{array} \right] \;=\; \tfrac{1}{2}(e_1 + e_2) - e_3 \\ {\vert 1 \rangle} \;\coloneqq\; \tfrac{\sqrt{3}}{2} \left[ \begin{array}{c} 1 \\ -1 \end{array} \right] \;=\; \tfrac{\sqrt{3}}{2}(e_1 - e_2) \,. \end{array}

On this bases the action of (213)(213) is

ρ(213)(|0)=|0 \rho(213)\big({\vert 0 \rangle}\big) \;=\; {\vert 0 \rangle}
ρ(213)(|1)=|1 \rho(213)\big({\vert 1 \rangle}\big) \;=\; -{\vert 1 \rangle}

and hence (213)(213) acts as the Pauli Z-gate:

ρ(213)=(1 0 0 1)=Z \rho(213) \;=\; \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right) \;=\; Z

On the other hand, the action of (132)(132) is found to be

ρ(132)(|0)=ρ(132)(12[1 1])=12[1 2]=12|0+32|1 \rho(132)\big({\vert 0 \rangle}\big) \;=\; \rho(132) \left( \tfrac{1}{2} \left[ \begin{array}{c} 1 \\ 1 \end{array} \right] \right) \;=\; \tfrac{1}{2} \left[ \begin{array}{c} 1 \\ -2 \end{array} \right] \;=\; -\tfrac{1}{2} {\vert 0 \rangle} + \tfrac{\sqrt{3}}{2} {\vert 1 \rangle}
ρ(132)(|1)=ρ(132)(32[1 1])=32[1 0]=32|0+12|1 \rho(132)\big({\vert 1 \rangle}\big) \;=\; \rho(132) \left( \tfrac{\sqrt{3}}{2} \left[ \begin{array}{c} 1 \\ -1 \end{array} \right] \right) \;=\; \tfrac{\sqrt{3}}{2} \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] \;=\; \tfrac{\sqrt{3}}{2} {\vert 0 \rangle} + \tfrac{1}{2} {\vert 1 \rangle}

and hence

ρ(132)=(1/2 3/2 3/2 1/2)=(1 0 0 1)(1/2 3/2 3/2 1/2)=(1 0 0 1)(cos(π/3) sin(π/3) sin(π/3) cos(π/3))=ZR y(2π/3) \rho(132) \;=\; \left( \begin{array}{c} -1/2 & \sqrt{3}/2 \\ \sqrt{3}/2 & 1/2 \end{array} \right) \;=\; \left( \begin{array}{c} -1 & 0 \\ 0 & 1 \end{array} \right) \left( \begin{array}{c} 1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2 & 1/2 \end{array} \right) \;=\; \left( \begin{array}{c} -1 & 0 \\ 0 & 1 \end{array} \right) \left( \begin{array}{c} cos(\pi/3) & -sin(\pi/3) \\ sin(\pi/3) & cos(\pi/3) \end{array} \right) \;=\; - Z \circ R_y(2\pi/3)


end

\lnebreak

The action of (231)(231)

(231)|0=12[2 1]=12|032|1 (231) {\vert 0 \rangle} \;=\; \tfrac{1}{2} \left[ \begin{array}{c} -2 \\ 1 \end{array} \right] \;=\; -\tfrac{1}{2} {\vert 0 \rangle} -\tfrac{\sqrt{3}}{2} {\vert 1 \rangle}
(231)|1=32[0 1]=32|012|1 (231) {\vert 1 \rangle} \;=\; \tfrac{\sqrt{3}}{2} \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] \;=\; \tfrac{\sqrt{3}}{2} {\vert 0 \rangle} - \tfrac{1}{2} {\vert 1 \rangle}

Hence

(231)=[1/2 3/2 3/2 1/2] (231) \;=\; \left[ \begin{array}{cc} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2 & - 1/2 \end{array} \right]

The action of (321)(321)

(321)|0=(321)12[1 1]=12[2 1]=12|032|1 (321) {\vert 0 \rangle} \;=\; (321) \tfrac{1}{2} \left[ \begin{array}{c} 1 \\ 1 \end{array} \right] \;=\; \tfrac{1}{2} \left[ \begin{array}{c} -2 \\ 1 \end{array} \right] \;=\; -\tfrac{1}{2} {\vert 0 \rangle} - \tfrac{\sqrt{3}}{2} {\vert 1 \rangle}
(321)|1=(321)32[1 1]=32[0 1]=32|012|1 (321) {\vert 1 \rangle} \;=\; (321) \tfrac{\sqrt{3}}{2} \left[ \begin{array}{c} 1 \\ -1 \end{array} \right] \;=\; \tfrac{\sqrt{3}}{2} \left[ \begin{array}{c} 0 \\ -1 \end{array} \right] \;=\; \tfrac{\sqrt{3}}{2} {\vert 0 \rangle} - \tfrac{1}{2} {\vert 1 \rangle}

Hence (321)(321) acts as

(321)=[1/2 3/2 3/2 1/2]=[1/2 3/2 3/2 1/2]=R y(2π/3) (321) \;=\; \left[ \begin{array}{c} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2 & -1/2 \end{array} \right] \;=\; - \left[ \begin{array}{c} 1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2 & 1/2 \end{array} \right] \;=\; - R_y(2\pi/3)



Defect anyons

Controlling anyons

Loop space of the 2-sphere

On the loop space ΩS 2\Omega S^2 of the 2-sphere

in relation to braid groups

  • Frederick R. Cohen, J. Wu: On Braid Groups, Free Groups, and the Loop Space of the 2-Sphere, in: Categorical Decomposition Techniques in Algebraic Topology, in Progress in Mathematics 215, Birkhäuser (2003) 93-105 [doi:10.1007/978-3-0348-7863-0_6]

On ΩS 2BΩ 2S 2\Omega S^2 \,\simeq\, B \Omega^2 S^2 regarded as a classifying space (for “l\mathbf{l}ine” bundles):

Last revised on March 17, 2025 at 10:45:13. See the history of this page for a list of all contributions to it.